The maximum-valency connectivity conjecture for Hom complexes

Let $G$ be a graph, let $d$ be its maximal valency, and let $K_n$ be the unlooped complete graph on $n$ vertices. For an integer $k\geq -1$, write $\text{Hom}(G,K_n)$ for the Hom complex and say it is $k$-connected in the usual topological sense. Maximum-valency connectivity conjecture. If the maximal valency of $G$ is $d$, then for all integers $k\geq -1$ and $n\geq d+k+2$, the complex

$$\text{Hom}(G,K_n)$$

is $k$-connected. The preceding proposition establishes the case $k=0$ under the weaker-looking threshold $n\geq d+2$. This conjecture proposes higher connectivity as the number of available colors increases; the supplied passage gives no resolution.